16 BAIRE'S THEOREM AND ITS CONSEQUENCES 93 11. Let E, F be two Frechet spaces, G a Banach space, and (/,) a sequence of continuous bilinear mappings of E x F into G. Suppose that, for each (x, y) e E x F, the se- quence (/„(#, y)) is bounded in G. If D is a dense subset of E x F and the sequence (fn(x, y)) converges for all (x, y) e D, show that the sequence (/n) converges simply in E x F and that the limit function is continuous. (Use Problem 10 and (7.5.5).) 12. Let E be the set of all sequences x = (£,)„=* 0 of real numbers such that the series with general term £„ is convergent, and put ||jc|| = sup (a) Show that ||#|| is a norm on E and that E is complete with respect to this norm. (b) The Banach space I1 (Section (5.7), Problem 1) is dense in E with respect to the 00 norm ||x||. If/1 is endowed with its norm ||jc||i = X lf»|, then the canonical injection n = 0 I1 -*E is continuous. (c) Let (Pn) be an increasing sequence of finite subsets of N x N, forming a covering of N X N. For each x = (£,) e E and each y = (rjn) e /*, let /„(*, y) = £ £ -n, . («. J)eP« Show that the sequence (/„) converges simply in E x I1 if and only if, for each (x, y) e E x /*, the sequence (fn(x,y)) is bounded, and that the limit of (/„(*, y)) is then equal to (ZfnHE7?") for a11 (*> jO s E x /*. (Use Problem 11 with D = I* x I1.) (d) For each y e N, let pjn be the smallest number of closed intervals in N whose union is the section Pn~ l(j) (the projection of P« n (N x {./'}))• Let pn = sup pjn , ,/eN Show that the condition in (c) is equivalent to suppn< +00. (If <pp/l is the it characteristic function of Pn , show that the norm of the bilinear form /„ (5.7.7) is equal to sup (E \<Prn(iJ)- 9»p.0'+ ./«N\«-0 ("Theorem of Mertens-Alexiewicz")* 13. The Banach space 71 (Section 5.7, Problem 1) is algebraically a vector subspace of the space /2 (6.5). If \\x\\i and \\x\\2 are the norms on I1 and I2 respectively, we have IMIa ^ ll^lli for all x e /*. Show that in the space /2, the unit ball B : ||x||i g 1 of I1 is not closed with respect to the norm ||^||2 , although the intersection of B with every finite-dimensional linear variety is closed (use (12.16.3)). 14. Let E be a Frechet space and F a normed space. If H is a set of continuous linear mappings of E into F which is not equicontinuous, show that the set of x e E such that H(x) (the set of all u(x) with u e H) is not bounded in F is the complement of a meager set. Deduce that if (Fw) is a sequence of Banach spaces and if, for each n, Hn is a subset of ^(E; Fn) which is not equicontinuous, then there exists XG eE such that none of the sets Hn(xQ) is bounded (u principle of condensation of singularities "). 15. Let (aij)t$j be a family of points of the interval I = [0, 1] in R. For each integer n ^ 0, n let cort(x) » TfT (x — «<„), and suppose that the roots of this polynomial are all distinct. Let (<iin)(x - atn))sfying the following conditions: (1) for